Abstract
An ordered compact space is a compact topological space X, endowed with a partially ordered relation, whose graph is a closed set of X × X (cf. [4]). An important subclass of these spaces is that of Priestley spaces, characterized by the following property: for every x, y ϵ X with x ≰ y there is an increasing clopen set A (i.e. A is closed and open and such that a ϵ A, a ⩽ z implies that z ϵ A) which separates x from y, i.e., x ϵ A and y ≱ A. It is known (cf. [5, 6]) that there is a dual equivalence between the category Ld01 of distributive lattices with least and greatest element and the category P of Priestley spaces. In this paper we shall prove that a lattice L ϵ Ld01 is complete if and only if the associated Priestley space X verifies the condition: (E0) D ⊆ X, D is increasing and open implies D ∗ is increasing clopen (where A ∗ denotes the least increasing set which includes A). This result generalizes a well-known characterization of complete Boolean algebras in terms of associated Stone spaces (see [2, Ch. III, Section 4, Lemma 1], for instance). We shall also prove that an ordered compact space that fulfils (E0) is necessarily a Priestley space.
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